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A monoidal Grothendieck construction for $\infty$-categories

Maxime Ramzi

TL;DR

This work develops a monoidal enhancement of the Grothendieck un/straightening for $\infty$-categories, showing that for any $\mathcal{O}$-monoidal $\infty$-category $\mathbf C$, the lax $\mathcal{O}$-monoidal functors $\mathbf C \to \mathbf{Cat}$ correspond to $\mathcal{O}$-monoidally coCartesian fibrations over $\mathbf C$, via a monoidal un/straightening equivalence. Building from a technical pullback lemma, the authors establish a microcosmic equivalence between lax monoidal functors and coCartesian fibrations, extend it to a metacosmic, Cat-linear equivalence that identifies the Day convolution on $\operatorname{Fun}(\mathbf C,\mathbf{Cat})$ with the monoidal structure on $\mathbf{coCart}_\mathbf C$, and finally describe the macrocosm in which $\mathbf{coCart}_\mathbf C$ sits as a non-full sub-operad of $(\mathbf{Cat}_{/\mathbf C})^\otimes$, with the un/straightening providing a symmetric monoidal equivalence. The paper also discusses the interaction with anima-valued functors, potential $(\infty,2)$-categorical refinements, and outlines a program to compare the macro- and micro-level pictures, highlighting both the core results and the remaining technical challenges.

Abstract

We construct a monoidal version of Lurie's un/straightening equivalence. In more detail, for any symmetric monoidal $\infty$-category $\mathbf C$, we endow the $\infty$-category of coCartesian fibrations over $\mathbf C$ with a (naturally defined) symmetric monoidal structure, and prove that it is equivalent the Day convolution monoidal structure on the $\infty$-category of functors from $\mathbf C$ to $\mathbf{Cat}_\infty$. In fact, we do this over any $\infty$-operad by categorifying this statement and thereby proving a stronger statement about the functors that assign to an $\infty$-category $\mathbf C$ its category of coCartesian fibrations on the one hand, and its category of functors to $\mathbf{Cat}_\infty$ on the other hand.

A monoidal Grothendieck construction for $\infty$-categories

TL;DR

This work develops a monoidal enhancement of the Grothendieck un/straightening for -categories, showing that for any -monoidal -category , the lax -monoidal functors correspond to -monoidally coCartesian fibrations over , via a monoidal un/straightening equivalence. Building from a technical pullback lemma, the authors establish a microcosmic equivalence between lax monoidal functors and coCartesian fibrations, extend it to a metacosmic, Cat-linear equivalence that identifies the Day convolution on with the monoidal structure on , and finally describe the macrocosm in which sits as a non-full sub-operad of , with the un/straightening providing a symmetric monoidal equivalence. The paper also discusses the interaction with anima-valued functors, potential -categorical refinements, and outlines a program to compare the macro- and micro-level pictures, highlighting both the core results and the remaining technical challenges.

Abstract

We construct a monoidal version of Lurie's un/straightening equivalence. In more detail, for any symmetric monoidal -category , we endow the -category of coCartesian fibrations over with a (naturally defined) symmetric monoidal structure, and prove that it is equivalent the Day convolution monoidal structure on the -category of functors from to . In fact, we do this over any -operad by categorifying this statement and thereby proving a stronger statement about the functors that assign to an -category its category of coCartesian fibrations on the one hand, and its category of functors to on the other hand.
Paper Structure (6 sections, 37 theorems, 30 equations)

This paper contains 6 sections, 37 theorems, 30 equations.

Table of Contents

  1. Pullbacks of monoidal structures
  2. The microcosmic monoidal Grothendieck construction
  3. The metacosmic monoidal Grothendieck construction
  4. The macrocosmic monoidal Grothendieck construction
  5. Comparison between macrocosmic and microcosmic versions
  6. Colimits of coCartesian fibrations

Key Result

Theorem 1

Let $\mathcal{O}$ be an $\infty$-operad and $\mathbf C$ an $\mathcal{O}$-monoidal $\infty$-category. There is an equivalence of $\infty$-categories between the $\infty$-category of lax $\mathcal{O}$-monoidal functors $\mathbf C\to \mathbf{Cat}$ and the $\infty$-category of $\mathcal{O}$- monoidally coCartesian fibrations over $\mathbf C$ which, on underlying objects is the un/straightening equiva

Theorems & Definitions (82)

  • Theorem 1
  • Theorem A
  • Theorem B
  • Remark 1
  • Corollary A
  • Corollary B
  • Definition 1
  • Remark 2
  • Example 1
  • Proposition 1
  • ...and 72 more